234 Xlll INTEGRATION 9. Let (Xn)w> o be an infinite sequence of compact spaces, and let pn be a positive measure on Xn with total mass 1, for each n. 00 (a) Show that on X = XI X« there exists a unique positive measure ^ such that, for each integer n and each finite sequence (/f)o^{<« of functions /, 6 ^R(Xi), we have ju(/) = II jLtf(/i), where /W = fl/t(Pr^)- (Observe that the continuous functions n of the form n/i(pri #), f°r a^ choices of n and of f{ e ^R(Xj), form a total set in the Banach space ^R(X).) The measure ju, is called the product of the family (/xn) and is n written p, = ® /xn • «=o (b) For each 71, let An be a /^-measurable subset of Xn. Show that the product 00 °0 A = FT An is u-measurable, and that ju(A) = Yi ^n(An). fi = 0 n=° (c) Let/^ 0 be a /x-integrable function on X. For each subset L of N, put L' = N — L, and identify X with the product XL x XL>, where XL = YL x« and XL> = Yl Xn. If n e L n e L' x 6 X, put xL = prLjc and xL' = prL<#, and identify x with (XL,XL,). Finally, let ju,L> be the product measure ® pa on XL', and put/L(#) = /(*L, *L') ^L- (*i/)- n € L' »/ Now let (Ln) be an increasing sequence of subsets of N, and put g = sup fLn and h = sup/Ln'. For each c > 0, let Ac be the set of points x e X at which g(x) > c, and Bc the set of points x e X at which h(x) > c. Show that c • /z(Ac) <M /d/x and c - ju(Bc) g I fdp,. (Remark that Ac is the set of points x e X for which at least one of the/Ln(x) is >c, and express Ac as a denumerable union of mutually disjoint sets Gn such that c • LL(G«) g f fdp,.) JG« (d) Suppose that (Ln) is an increasing sequence of finite subsets of N whose union is N. Show that the/Ln tends to/almost everywhere and that/Ln tends to the constant fdfjb almost everywhere. (For each e > 0, consider a continuous function #, depend- ing on only finitely many variables, and such that \f—g\ dyi :ge, and apply (c) to the function |/— g\.) 10. Let D be the discrete space {0, 1} and X the product space DN. On each factor Xn (= D) of X let un be the measure for which ju,n(0) = /^n(l) = i, and let a = ® un be the n e N product measure on X (Problem 9). 00 (a) For each point x = (xn)n>0 in X with xn = 0 or xn = 1, put ir(x) ~ XI ^«2"ir~I. n = 0 Show that 77 is a continuous mapping of X onto the unit interval I — [0, 1 ] in R, and that the image TT(IJL) of /x is Lebesgue measure on I. For each /el, the set TT"!(/) consists of a single point, unless / is of the form k - 2"" with k integral and 0 < k < 2". Hence the mapping /h-*/° TT induces, on passing to the quotients, an isometric isomorphism of L£(I, A) onto L£(X, JLC) for 1 <p g + oo. (b) For each / e J and each « ^ 1, put rn(/) — 1 — 2prn_i(7r~1(0) if t is not of the form k * 2~m with k integral and 0 g k < 2m, and rn(0 = 0 otherwise. The function rn is called the nth Rademacher function. These functions form a nontotal orthonormalnuous on the right, defined by w(t) — u(t)v(t) for / J> 0 and w(r) = 0 for / < 0.